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# Proof of Main Theorem

Lemma 2   : Let be a square-free integer, be the greatest integer in , as defined in (4) and . Then if and only if

Proof: For a proof, see [??].

Theorem 3 (B = 2C)   : Let be a square-free integer such that the continued fraction representation of has a center b, be the greatest integer in , , and be defined as in (4). Then if and only if .

Proof: Suppose that . Then so by Lemma 2,

Carrying out this fractional linear transformation yields

Substituting in for and making the substitution stated in (8) gives the equation

But by the construction of , det =. Thus making the substitution into the above equation, we get the quadratic equation (in B)

Solving for yields

 (9)

But by the choice of and the construction of and , we have that and thus the second possibility must be ruled out because it would yield a negative value for . Hence as required. Conversly, assume that . Then it is true that

But once again, so if we make the substitution , it is true that

Then making the substitution and the one stated in (8) yields (after rearranging terms)

It follows that

and thus by Lemma 2, and .

For the remainder of the report we consider those continued fractions such that . Then we can make a change of variables in the matrix in (4) and let

 (10)

Next: Factorization Up: Continued Fraction Factorization Previous: Theorems of Special Interest

scanez 2000-12-04